(Some of the following games are derived from my local folk culture.
Others are in Macfarlans' books.)
The various forms of Indian wrestling may a formed into a tournement.
These include:

LEG -

lying hip to hip heads opposite, the opponents raise the inside leg
and on the count of three catch the opponents leg and attempt to turn the
opponent over, loser is the one turned,

ARM -

seated facing, elbows together on table, Hands grasped, opponents attempt
to force the others hand to the table wihtout lifting elbow from table.
Loser is one whose hand is to the table

STAFF (fixed feet) -

Standing facing with wide stance, both opponents grasp with both hands
(right by left and left by right) a stout staff, on the signal they attempt
to without letting go of the staff or moving their feet force their opponent
to let go of the staff or move his feet.

STAFF (nonfixed feet) -

Standing facing with wide stance, both opponents grasp with both hands
(right by left and left by right) a stout staff, on the signal they attempt
to without letting go of the staff, force their opponent to let go of the
staff

STAFF (prone):

The opponents lay on their stomachs on the ground facing each other
so distanced and placed that their left elbows touch, feet straight out
and together (for rougher game, each may spread their feet). A 24"
x 2" diameter staff is placed in their hands such that each left hand grabs
the staff near the center and the right hands grasps the staff near the
opposite ends. The opponents hands will be r(1) - l(2) - center -
l(1) - r(2), i.e. alternate on the staff. At the ready command,
each prepares, at the start command, each attempts to roll his opponent
over, without changing his own position. Wiles and Bluff are important
in this.

THUMB -

opponents grasp the opponents hand (right to right or left to left)
so that their curved fingers interlock and the thumbs can easily reach
each other. On the third switch back and forth, they attempt to get
their thumb on top and hold the opponents thumb down for a five count.

BALANCE -

opponents stand facing, slightly offset so their right feet can be
placed forward and side to side, they grasp each others right wrists and
push and pull, first player to move forward placed foot loses. (may
also be played left footed and handed)

HOP AND PULL:

opponents stand on their right foot, grasp right hands. at the
start command, each attempts to pull the other off balance. Left
foot may not touch the ground, right foot may move hop fashion. (may also
be played left footed and handed.)

SIT WRESTLE:

Two challengers sit back to back with elbows locked. On the start
signal, they each attempt to force the others right should to the ground.
The game is played two out of three for the winner. (Macfarlan p
233)

BACK (arms behind):

Two opponents stand chest to chest with hands clasped (or tied)
behind their backs. Pushing only with their chests, each attempts
to push the other cross a line 5 feet behind him.

BACK TO BACK (standing):

Opponents stand back to back, locking elbows. Each attempts to
pull, tug, carry or otherwise get his opponent cross a line 5 feet in front
of himself.

BACK TO BACK: (crab):

Opponents sit back to back. On the ready signal they raise themselves
up on their hands so as to be in crab position (supported on hands and
feet only, face up). At the start command each attempts to push his opponent
cross a line 10' behind him.

BACK (arms outstretched):

Two challengers stand chest to chest with arms outstretched.
About two paces behind each is a line drawn in the dirt. At "ready",
the challengers firmly push chest to chest. On "push" they attempt
to push each other over the line behind him. This is decided on a
two out of three. (Macfarlan p 235)

PUSH:

Oponents stand back to back, arms folded in front. Each attempts
to push the other back cross a line 6 feet behind him.

BALANCE AND PUSH:

Two opponents stand on facing each other, each standing only on his
right foot. Their left feet are opposed sole to sole. On start,
they attempt to push or maneuver the other so as to cause the other to
fall, or touch his left foot to the ground. No kicking, no jabbing,
only steady pushing.

FOOT PULL:

Oppenents sit facing each other, legs parallel, inside foot at opponents
inside knee. At the ready command, each grabs a hold of his opponents
near foot, at the pull command, each attempts to pull the other's foot
to his waist. The leg is pulled straight without raising it above
the level of the flat legs.

STRONG HAND:

The opponents kneel facing each other, at the ready command they lock
fingers palm to palm, left to right and right to left. At the push
command, they attempt to force their opponents shoulders to the ground,
in either direction. Only the hand contact is allowed.

LIFT (structured):

The contestants stand facing each other. Each places his right
arm under the opponents left arm and around his back. The left arm
is placed over the opponents right shoulder. At the start command,
each attempts to lift the other off the ground. The first lifted
loses.

LIFT (unstructured):

The contestants stand facing each other. At the start command, each
attempts to lift the other off the ground. The first with both feet
off the ground loses.

ARM PULL:

Two opponents are placed back to back. Each bends forward and
extends his right arm between his legs, grasping the wrist of his opponent.
At the start each attempts to pull the other cross a line 10' in front
of himself.
VARIATION: A set of amount of time (10 to 20 sec) is allowed.
The one pulling his opponent furthest wins.

REACH:

Two opponents grasp either right or left wrists. A small stick
or other object is placed 6 feet behind each. On the signal, each
attempts to grab the object behind himself, without releasing the locked
wrist grasp. First to obtain his object wins. Usually played
2 out of three.

COUP STICK:

A stout hardwood pole about 4 feet long is grasped by two opponents
facing each other, with both hands in such fashion that the right hands
almost touch at the middle and their left hands are about a foot wider.
At the start signal they attempt to take the stick away from the other.
The first to lose grasp with both hands loses. One hand or the other
may be quickly adjusted or replaced, but not both at once. (the game
may be softened up some by requiring that the feet be kept wide placed.)
(Macfarlan p 231)

BULLING:

The opponents face each other. Bending forward, each places his
right arm behind the others neck, and his left hand on hip opponents
right elbow. At the start command, each attempts to force the other
to move a foot. First to move a foot loses.

WESTMORELAND WRESTLING:

The opponents face each other. Bending forward, each places his
right arm behind the others neck, and his left hand on hip opponents
right elbow. At the start command, each attempt to force his opponent
to touch the ground with any part of his body other than his feet.
First to touch the ground with other than his feet loses. If grip
is lost, the match restarts.